Rules for Manipulating Summations
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Theorem
Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ and $S: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be propositional functions on the set of integers.
Let $S: \Z \times \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be a propositional functions on the Cartesian product of the set of integers with itself.
Let $\displaystyle \sum_{R \left({i}\right)} x_i$ denote a summation over $R$.
Let $\pi$ be a permutation on the fiber of truth of $R$.
Change of Index Variable of Summation
- $\displaystyle \sum_{R \left({i}\right)} a_i = \sum_{R \left({j}\right)} a_j$
Permutation of Indices of Summation
- $\displaystyle \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$
General Distributivity Theorem
- $\ds \paren {\sum_{i \mathop = 1}^m a_i} * \paren {\sum_{j \mathop = 1}^n b_j} = \sum_{ {1 \mathop \le i \mathop \le m} \atop {1 \mathop \le j \mathop \le n} } \paren {a_i * b_j}$
Exchange of Order of Summation
- $\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} a_{i j} = \sum_{S \left({j}\right)} \sum_{R \left({i}\right)} a_{i j}$
Exchange of Order of Summation with Dependency on Both Indices
- $\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({i, j}\right)} a_{i j} = \sum_{S' \left({j}\right)} \sum_{R' \left({i, j}\right)} a_{i j}$
where:
- $S' \left({j}\right)$ denotes the propositional function:
- there exists an $i$ such that both $R \left({i}\right)$ and $S \left({i, j}\right)$ hold
- $R' \left({i, j}\right)$ denotes the propositional function:
- both $R \left({i}\right)$ and $S \left({i, j}\right)$ hold.
Sum of Summations over Overlapping Domains
- $\displaystyle \sum_{R \left({j}\right)} a_j + \sum_{S \left({j}\right)} a_j = \sum_{R \left({j}\right) \mathop \lor S \left({j}\right)} a_j + \sum_{R \left({j}\right) \mathop \land S \left({j}\right)} a_j$
where $\lor$ and $\land$ signify logical disjunction and logical conjunction respectively.